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Tsiolkovski Factor.


Tsiolkovski Parameter   V2 = Vin*Ln(M1/M2) m/s


Where Vin is the velocity of the fluid/gas jet stream leaving the rocket nozzle m/s 

M1 Initial fueled mass of the rocket. Kgs.

M2 Final mass of the rocket . Kgs.

V2  Is the actual final velocity of the rocket with mass M2 m/s

Ln  Natural Logarithm.




Looking at the formula for determining the final velocity of the rocket.

We can say that if the ratio M1/ M2 is greater than 2.7182 then the rate of mass change will accelerate the final rocket mass M2.

For practical examples of this law refer to the Tsiolkovski Formula application table.

Ariane family courtesy esa 


  • To optimise the acceleration of the rocket the ratio of initial mass M1 to final mass M2 needs to be as large as possible.
  • The rate of change of mass needs to be optimum. For this to occur there needs to be maximum mass flow-rate.
  • The final performance of the rocket will be improved  by removing the parts that have finished their useful life as early in the flight as possible. So reducing the mass of the rocket to be accelerated.


Multiple stage rockets will accelerate their final mass M2 more efficiently.

In practice this is achieved by jettisoning the initial launch booster rockets some 2 mins into the flight. When they have burnt all their fuel.

Similarly each stage is removed once it has completed its task. Reducing the value of M2.

First stage, second stage etc.

This was predicted by Tsiolkovski on 10th May , 1897 and published in1903 . Over the years the theory developed with the definitive version being published in 1929.in a classic work written during the last 10 years of his life entitled 'Space Rocket trains'.

Tsiolkovslki understood the importance of multi-staging not just of modules assembled in series piled one on top of the other but also using parallel modules. 

A good example of the rocket modules piled high in series is the US Saturn V  launcher used for the Apollo moon missions. Today rocket engineers would not follow the same design philosophy. Preferring that of using parallel modules that can be ejected earlier in the rocket flight. Note: Taller is not always better.

For a fully worked example refer to mit:

Advanced fluid dynamics course online



New Water Rocket Explorer derivation of Tsiolkovski formula from Propulsion Force derivation PDF

Propulsion force formula summary with Tsiolkovski derivationPDF

My derivation of Tsiolkovski's Rocket Formula  PDF

New My excel spreadsheet for the application of Tsiolkovski's formula to multi-stage rockets 1924

Application of Tsiolkovski formula to 2stage rockets PDF

This derivation of the Tsiolkovski formula linked below needs to be expressed more clearly.




Water rockets and Tsiolkovski


V2 = Vin *Ln(M1/M2) m/s


Water rockets are accelerated by using a small volume of water in the same way.

Note: Vin for water rockets can be assumed to be equal to the initial rocket velocity at the bottom of the water spike. As water  leaves the rocket in a jet.  Vin is equal to the velocity of the water jet stream leaving the nozzle.

Because the event time of the water rocket is relatively short ( In the order of 0.04>0.08seconds )Then we could assume an analogy where the water jet provides an impulse to the rocket.

During this impulse the rocket mass  Mreduces, accelerating the rocket towards the top of the water spike.

The time it takes for this mass change from M1 to M2 to occur, also determines the value of the jet velocity produced. 

It is the rate of loss of mass between the initial M1 and the final rocket mass M2 that is important.

Flight Test data.

During the initial thrust phase of a 7bar  launch( which lasts for only 0.06s). The rocket experiences an acceleration of between 30 to 40 g. 


Note: Using high speed video we have recorded values in excess of 4000m/s2 at 8bar with a basic Badoit 1liter rocket.

So momentarily we can assume that the water rocket is exposed to an extreme acceleration. 

Which will introduce severe acceleration and deceleration loads on the projectile/rocket.


Definition of Water spike

The 'water spike' is the water jet stream created by a water rocket after launch. During the initial impulse. Difficult to see with the naked eye this water spike can be seen by slowing a launch video using VirtualDub.

A comparison of multistage an single stage Water Rocket performance.Using Tsiolkovski  Formula Excel

Comparison of rate of change in mass ratio with Ariane 5.

Physics experiment to confirm Tsiolkovski's formula using a filmed water rocket launch.

Note: That the full Tsiolkovski formula as applied to real rockets is an interactive formula that calculates the actual velocity of the rocket adjusted for the change in mass over a fixed time period t

 V2 = Vo +Vjet*Ln(Mo/M2) m/s


Where Vo = Initial rocket velocity m/s

            Mo = The original mass with velocity Vo.

            Vjet = Velocity of flow at the outlet of the nozzle.m/s

             V2  = Corrected velocity of the rocket projectile after interval t.secs


This approach has been used in the duct flow formula model spreadsheet. Where the modified water rocket velocity has been calculated over a number of incremental time steps. That represent the water jet impulse phase.


The importance of Specific Impulse

The more familiar delta V form of the equation      is used for calculating the change in velocity require to either escape the surface of a planet , attain a required orbit or that required to voyage between two planets.

Delta V Budget for LEO transfer to GEO



Escape velocities and Mach Number


The Mach number relates the vehicle velocity to that of the velocity of sound in air (340m/s or1200Km/hr).

Commercial aircraft attain a velocity of 0.8Mach (900Km/hr),whilst supersonic aircraft like Concorde travel at Mach 2 ( 2400Km/hr). The Russian Mig 25 (Foxbat) reaches speeds in excess of 3000 Km/hr.

This compares with rockets that attain speeds of the order of Mach23 (27600 Km/hr or 7.82Km/s) in order to place a satellite in a base orbit of 200kms. This is known as the First Cosmic escape velocity.

Second Cosmic velocity 11.2Km/s is the velocity required to escape the attraction of the earth.

Named after the Austro-German physicist and philosopher Ernst Mach (1838-1916) who identified the relationship of the speed of sound to the aerodynamic behaviour of bodies travelling through air.


Rocket Escape velocities

First cosmic velocity     =    7.82Km/sec

Second cosmic velocity = 11.2  Km/sec

Third cosmic velocity    =  16.6  Km/sec


As a comparison , some of the better water rockets travel at over100m/s or 0.3Mach.


Ernst Mach is also responsible for a publication in 1883 which postulated that the laws of relativity in Newtonian physics and mechanics should take into account  the effects of the universe or cosmos in which they occur. This idea interested Einstein who produced his first theory on relativity in 1905. Subsequently developing the idea  to take account of Mach's hypothesis producing the 'General theory of relativity' of 1915 in which he broadened the scope of the initial theory to include the effects of the cosmos. Followed in 1917 by his first cosmic theory on the universe  to which Einstein dedicated the title 'Principe de Mach'

Mach's principle: The inertial forces experienced by a body in non-uniform motion are determined by the quantity and distribution of matter in the Universe


Calculating Escape velocty


soyouz.jpg (71977 bytes)

Soyouz courtesy esa 



Video Soyouz 11 Manned flight


Starsem selection of Soyuz launch videos.



Soyuz Fregate launch video of the Galaxy14  mission.Fantastic video sequence showing night launch and the first and second stage seperation.

Duration 9 mins. First stage seperation 3.52mins video synch. High atmospheric plume due to low external pressure outside the motor expansion nozzle 5.2mins.



Arianespace link to launch videos Ariane 5.



 Soyuz Rocket specification

Sutton, George Paul. Rocket Propulsion Elements: An Introduction to the Engineering of Rockets, 6th ed. New York: Wiley, 1992. 636.


In honour of Ernst Mach here is a link to the sound track by Cold Play 'Speed of Sound'



Based on Arianespace A5 Flight data A167pdf


Real Rockets



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This site was created on the 15th April 2003

ŠJohn Gwynn and sons2003 

You're welcome to reproduce any material on this site for educational or other non commercial purposes

 as long as you give us proper credit (by referring to "The Water-Rocket Explorer" http://waterocket.explorer.free.fr).